Question

ere Cesaro Sumrnability. Given an infinite series Σ an let Sn be the sequence of partial sums and let 5 Tt A series is Cesaro-surmable if linn-troƠn exists (and is finite). and this limit is called the Cesàro sum (a) Given the series 2n-1 n' s", hnd 8m and Ơn for any 1. and find the Cesaro sum of ΣΥ_1)". (b) Find the Cesàro sum of Here you may use the fact, discussed in class, that cos(2k 1)r sin 2n.r 2 sin r

Please write it clearly and show every step

Answer 1

S s t partial eums and v depe tt ntop) exists -then coe say that be sealer Tigt Cesaro Semomable let Sn be sequence of partial sums of Σ a then 2C1-2) lt 的che given Serie斥 Cost2kDX and idenbity given a k디 의 let Sn be sequence of partial sum Sin 2nx 2 Sina わ

2. sPn 22 わ25m 2 Sirz as Stm Dz and Sin are bounded finchoms